## Caption

## Extracts from the Article What's this?

This surface is shown in the (ω _{A} ,ω _{η}
,Ro)-space in Fig. 5(a) and in projection to the (ω
_{A} ,ω _{η} )-plane in Fig. 5(b) for
Re=1.

Indeed, for a given Lu, tending the magnetic field to zero along
a ray ω _{A} =ω _{η} Lu in the (ω _{A} ,ω
_{η} )-plane results in a value of the Rossby number
specified by (39), see Fig. 5(c).

Figure 5(d) demonstrates transition between the cases of high conductivity (Velikhov 1959) and of low conductivity (Chandrasekhar 1953) separated by the threshold $\omega_{\eta}= ({\omega_{\nu}^{2}+4\varOmega_{0}^{2}\alpha^{2}})(2\omega_{\nu})^{-1}$ .

Figure 5(d) also illustrates the conclusions of Acheson and
Hide that in the presence of small but finite resistivity in the
limit of vanishing ω _{A} “the stability or otherwise of
the system will then be decided essentially by Rayleigh’s
criterion” [22]..

This scenario corresponds to the lower Whitney umbrella singularity on the SMRI threshold surface shown in Fig. 5(a).

In the absence of viscosity and resistivity the roots of the
dispersion equation (23) corresponding to slow and fast MC-waves
are exactly 41 $$
\gamma^2=-2\varOmega_0^2\alpha^2(1+\mathrm{Ro})-\omega_A^2 \pm
2\varOmega_0\alpha\sqrt{\varOmega_0^2\alpha^2(1+\mathrm{Ro})^2+\omega_A^2}.$$
The corresponding double zero eigenvalue at ω _{A} =0 and
Ro=0 is related to the upper Whitney umbrella singularity at the
threshold surface of SMRI in Fig. 5(a)..

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